Ph.D. Defense: Jonathan Williams
Ph.D. Thesis Defense
Friday, April 5th, 2019
125 Hanes Hall
Nonpenalized model selection via generalized fiducial inference and Bayesian hidden Markov models
This dissertation is comprised predominantly of two topics of research. One involves extensive theoretical work on nonpenalized model selection problems encompassing both the high-dimensional linear regression and vector autoregression setting. The second is an application of a hidden Markov model (HMM) to the study of aging and the progression of dementia. Significant contributions of these projects are the implementation of alternative frameworks for statistical inference, namely, generalized fiducial inference (GFI) and Bayesian inference.
On the first topic, standard penalized methods of variable selection and parameter estimation rely on the magnitude of coefficient estimates to decide which variables to include in the final model. However, coefficient estimates are unreliable when the design matrix is collinear. To overcome this challenge an entirely new perspective on model selection is presented within a generalized fiducial inference framework. This new procedure is able to effectively account for linear dependencies among subsets of covariates in a high-dimensional setting where p can grow almost exponentially in n. Furthermore, with a typical sparsity assumption, it is shown that the proposed method is consistent in the sense that the probability of the true sparse subset of covariates converges in probability to 1 for large n and/or p. The model selection methodology is also extended from the linear regression setting to the vector autoregressive setting. In the extension, we construct methodology via the epsilon-admissible subsets (EAS) approach for posterior-like inference of relative model probabilities over all sets of active/inactive components of the VAR transition matrix. We provide a mathematical proof of strong graphical selection consistency for the EAS approach for stable VAR(1) models, and demonstrate numerically that it is an effective strategy in high-dimensional settings.